Page 196 - Applied Numerical Methods Using MATLAB

FALSE POSITION OR REGULA FALSI METHOD 185
x 4
k a k x k b k f(x ) 2.0375
k
5
0 1.6 3.0 32.6, −2.86 x x 3
1 1.6 2.3 3.0 −1.1808 2 2.125 x
2 1.6 1.95 2.3 0.5595 1.95 1
2.3
3 1.95 2.125 2.3 −0.5092 0
4 1.95 2.0375 2.125 −0.5027
. . . . .
. . . . .
. . . . .
1.6 1.8 2 2.2 2.4 2.6 2.8
(a) Process of the bisection method (b) The graph of f(x) = tan(p − x) − x
Figure 4.2 Bisection method for Example 4.2.

4.3 FALSE POSITION OR REGULA FALSI METHOD

Similarly to the bisection method, the false position or regula falsi method starts
with the initial solution interval [a, b] that is believed to contain the solution of
f(x) = 0. Approximating the curve of f(x) on [a, b] by a straight line connecting
the two points (a, f (a)) and (b, f (b)), it guesses that the solution may be the
point at which the straight line crosses the x axis:

f(a) f(b) af (b) − bf (a)
x = a − (b − a) = b − (b − a) =
f(a) − f(b) f(b) − f(a) f(a) − f(b)
(4.3.1)

function [x,err,xx] = falsp(f,a,b,TolX,MaxIter)
%bisct.m to solve f(x)=0 by using the false position method.
%input : f = ftn to be given as a string ’f’ if defined in an M-file
% a/b = initial left/right point of the solution interval
% TolX = upperbound of error(max(|x(k)-a|,|b-x(k)|))
% MaxIter = maximum # of iterations
%output: x = point which the algorithm has reached
% err = max(x(last)-a|,|b-x(last)|)
% xx = history of x
TolFun = eps; fa = feval(f,a); fb=feval(f,b);
if fa*fb > 0, error(’We must have f(a)f(b)<0!’); end
fork=1: MaxIter
xx(k) = (a*fb-b*fa)/(fb-fa); %Eq.(4.3.1)
fx = feval(f,xx(k));
err = max(abs(xx(k) - a),abs(b - xx(k)));
if abs(fx) < TolFun | err<TolX, break;
elseif fx*fa > 0, a = xx(k); fa = fx;
else b = xx(k); fb = fx;
end
end
x = xx(k);
if k == MaxIter, fprintf(’The best in %d iterations\n’,MaxIter), end